Syllabus
* Fundamentals of the theory of finite and Lie groups
Groups and their subgroups (basic properties and theorems), group homomorphism and isomorphism, group action on a set, Lie groups and its algebra (geometrical and matrix approach), one-parameter subgroups of the Lie group and exponential map, summary of matrix groups and their properties (double cover of SO(3) by SU(2))
* Fundamentals of the representation theory of groups
Representation as a group action on linear spaces, invariant subspaces, equivalent, unitary, irreducible, and (completely) reducible representations, basic theorems for finite and compact Lie groups (Schur's lemma, orthogonality relations, characters and their properties, Peter-Weyl theorem, Casimir operators, Racah theorem), summary of results of the representation theory of the symmetric group and the group SU(n)
* Applications in quantum theory
Classification of eigenvalues and eigenstates of an operator by irreducible representations of a symmetry group, coupled systems and decomposition of reducible representations (Clebch-Gordan series and coefficients), evaluation of matrix elements using group-theoretical methods (irreducible tensor operators, general Wigner-Eckart theorem, selection rules)
All notions and theorems will be illustrated by examples of point groups (which describe molecular and crystal symmetries and which play important role in quantum chemistry, molecular spectroscopy and solid state physics) and selected Lie groups such as SO(3), SU(2), and SU(3) (which are important in atomic, nuclear and particle physics).
Previous knowledge of groups is not assumed, but working knowledge of linear algebra and basic quantum mechanics is necessary for application.
Annotation
In this course, students become familiar with basic notions and results of the group theory and the representation theory for both finite and continuous (Lie) groups and learn how to use them to solve problems in physics. For the 1st and 2nd year of the TF and JSF studies.